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Explanation of the Gibbs Paradox within the Framework of Quantum Thermodynamics. Theo M. Nieuwenhuizen . Physikalisches Kolloquium Johann Wolfgang Goete Universitaet Frankfurt am Main 31-01, 2007. Outline.

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explanation of the gibbs paradox within the framework of quantum thermodynamics
Explanation ofthe Gibbs Paradoxwithin the Framework ofQuantum Thermodynamics

Theo M. Nieuwenhuizen

Physikalisches KolloquiumJohann Wolfgang Goete Universitaet

Frankfurt am Main

31-01, 2007


Who was Josiah Willard Gibbs?What is the Gibbs Paradox?On previous explanations: mixing entropy

Crash course in Quantum Thermodynamics

Maximal work = ergotropy

Application of mixing ergotropy to the paradox


Josiah Willard Gibbs

1839 – 1903

Carreer in Yale

1866-69: Travel to Paris, Berlin, Heidelberg

Gustav Kirchhoff, Hermann von Helmholtz

Gibbs free energy

Gibbs entropyGibbs ensemblesGibbs Duhem relation

Gibbs distribution

Gibbs state

Gibbs paradox

Copley Medal 1901


The Gibbs Paradox (mixing of two gases)Josiah Willard Gibbs 1876

mixing entropy

But if A and B identical, no increase.

The paradox: There is a discontinuity, still k ln 2 for very similar but non-identical gases.


Gibbs --------------------------------------------------------------------


= N log 2

proper setup for the limit b to a
Proper setup for the limit B to A
  • Isotopes: too few to yield a good limit
  • Let gases A and B both have translational modes at equilibrium at temperature T,but their internal states (e.g. spin) be described by a different density matrix or Then the limit B to A can be taken continuously.
current opinions
Current opinions:

The paradox is solved within information theoretic approach to classical thermodynamics

Solution has been achieved within quantum statistical physics due to feature of partial distinguishability

Quantum physics is right starting point.But due to non-commutivity, the paradox is still unexplained.

quantum mixing entropy argument
Quantum mixing entropy argument

Von Neuman entropy

After mixing

Mixing entropy

ranges continuously from 2N ln 2 (orthogonal) to 0 (identical) .Many scholars believe this solves the paradox.

Dieks & van Dijk ’88: thermodynamic inconsistency, because there is no way to close the cycle by unmixing.If nonorthogonal to any attempt to unmix (measurement) will alter the states.

another objection lack of operationality
Another objection: lack of operationality
  • The employed notion of ``difference between gases’’ does not have a clear operational meaning.
  • If the above explanation would hold, certain measurements would not expose a difference between the gasses. So the ``solution’’ would depend on the quality of the apparatus.
  • There is something unsatisfactory with entropy itself. It is non-unique. Its definition depends on the formulation of the second law.
  • To be operational, the Gibbs paradox should be formulated in terms of work.Classically: . . Also in quantum situation??
quantum thermodynamics thermodynamics applying to
Quantum Thermodynamics= Thermodynamics applying to:
  • System finite (small, non-extensive)
  • Bath extensive
  • Work source extensive (e.g. laser)

No thermodynamic limit

Bath has to be described explicitlyNon-negligible interaction energy

caldeira leggett model particle harmonic bath
Caldeira-Leggett model: particle + harmonic bath

Langevin equation (if initially no correlation between S and B)

first law is there a thermodynamic description though the system is finite
First law: is there a thermodynamic description,though the system is finite?

where H is that part of the total Hamiltonian,

that governs the unitary part of (Langevin) dynamicsin the small Hilbert space of the system.

Work: Energy-without-entropy added to the system bya macroscopic source.

1) Just energy increase of work source2) Gibbs-Planck: energy of macroscopic degree of freedom.

Energy related to uncontrollable degrees of freedom

Picture developed by Allahverdyan, Balian, Nieuwenhuizen ’00 -’04

roger balian 1933 cea saclay academie des sciences
Roger Balian (1933-)CEA Saclay; Academie des Sciences

B phase =Balian –Werthamer phase

(p-wave pairing)

- Eigenfrequencies of Schroedinger operators in finite domain

- Casimir effect: Balian-Duplantier sum rule

- Book: From microphysics to macrophysics- Quantum measurement process

second law for finite quantum systems
Second law for finite quantum systems

No thermodynamic limit Thermodynamics endangered

Different formulations are inequivalent

-Generalized Thomson formulation is valid:

Cyclic changes on system in Gibbs equilibrium cannot yield work

(Pusz+Woronowicz ’78, Lenard’78, A+N ’02.)

  • Clausius inequality may be violated
  • due to formation of cloud of bath modes

- Rate of energy dispersion may be negative Classically: = T* ( rate of entropy production ): non-negative

A+N: PRL 00 ; PRE 02, PRB 02, J. Phys A 02

Experiments proposed for mesoscopic circuits and quantum optics.


Armen Allahverdyan

Yerevan, Armenia

statistical mechanicsquantum thermodynamicsquantum measurement process

astrophysics, cosmology, arrow of time

adiabatic theoremsquantum optics

quantum work fluctuations

Gibbs paradox

> 35 common papers

work extraction from finite q systems
Work extraction from finite Q-systems

Couple to work source and do all possible work extractions

Thermodynamics: minimize final energy at fixed entropyAssume final state is gibbsian: fix final T from S = const.Extracted work W = U(0)-U(final)

But: Quantum mechanics is unitary,

So all n eigenvalues conserved: n-1 constraints, not 1.

(Gibbs state typically unattainable for n>2)

Optimal final situation: eigenvectors of become those of H

maximal work ergotropy
Maximal work = ergotropy

Lowest final energy:highest occupation in ground state,one-but-highest in first excited state, etc(ordering )

Maximal work:

(divine action, Aristotle)

Allahverdyan, Balian, Nieuwenhuizen, EPL 03.

aspects of ergotropy

-non-gibbsian states can be passive

Aspects of ergotropy

-Comparison of activities:

Thermodynamic upper bounds: more work possible from

But actual work may be largest from

- Coupling to an auxiliary system : if is less active than

Then can be more active than

-Thermodynamic regime reduced to states that majorize one another

- Optimal unitary transformations U(t) do yield, in examples, explicit Hamiltonians for achieving optimal work extraction

resolution of gibbs paradox
Resolution of Gibbs paradox
  • Formulate problem in terms of work:mixing ergotropy = maximal extractable work before mixing – ( idem, after mixing)
  • Consequence: limit B to A well behaved: vanishing mixing ergotropyParadox explained.
  • Operationality: difference between A and B depends on apparatus: extracted work need not be maximal
  • More mixing does not imply more work, and vice versa.Counterexamples given in A+N, PRE 06.

Luca Leuzzi, RomePhD in Amsterdam 2002, cum laude

L+N book: Thermodynamics of the glassy state


Gibbs paradox not solved up to nowMixing entropy argument has its own drawbacks

Explanation by formulation in terms of workMixing ergotropy = loss of maximal extractable work due to mixing

Operational definition: less work from less good apparatus

More mixing does not imply more work and vice versaMany details in Allahverdyan + N, Phys. Rev. E 73, 056120 (2006)


Are adiabatic processes always optimal?

One of the formulations of the second law:

Adiabatic thermally isolated processes done on an equilibrium system are optimal (cost least work or yield most work)

In finite Q-systems: Work larger or equal to free energy difference

But adiabatic work is not free energy difference.

A+N, PRE 2003:

-No level crossing : adiabatic theorem holds

-Level crossing: solve using adiabatic perturbation theory.

Diabatic processes are less costly than adiabatic.Work = new tool to test level crossing.

Level crossing possible if two or more parameters are changed. Review expts on level crossing: Yarkony, Rev Mod Phys 1996